Cosmology Eta Bprefactor Derivation Two Sided Corrected Lt One Sided

The two-sided correction

The number of protons and neutrons left over from the early universe is not zero. For every billion photons, about six baryons survive, a ratio measured by the Planck satellite as (6.10 ± 0.04) × 10⁻¹⁰. Recognition Science (RS) works from a discrete ledger, a record of recognition events, and in that framework the leading prediction for this ratio is φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰, about 4.5 percent above the central measured value.

The gap is small but real. A first-order correction multiplies the leading term by (1 − φ⁻⁸), where φ is the golden ratio, giving about 6.24 × 10⁻¹⁰, still 2.3 percent high. The declaration two_sided_corrected_lt_one_sided proves that squaring that same factor, (1 − φ⁻⁸)², produces a smaller value. The squared prefactor c_RS ≈ 0.9579 times φ⁻⁴⁴ lands at about 6.11 × 10⁻¹⁰, inside the 1σ error band of the Planck measurement. The theorem certifies the inequality: the two-sided correction is strictly less than the one-sided one.

The square has a natural reading. If matter and antimatter each suffer an independent washout, one factor of (1 − φ⁻⁸) per sector, the product of the two survival probabilities is the square. That interpretation is a hypothesis, not a theorem. The machine-checked library of formal theorems proves the algebra and the numerical bounds: c_RS lies strictly between 0 and 1, the corrected value lies in the band (6.0, 6.2) × 10⁻¹⁰, and that band contains the measured central value. The dynamical derivation from a Boltzmann equation, with the usual sphaleron and thermal factors, remains open.

The honest verdict is that the corrected number agrees with observation, but the reason for the square is not yet derived. The falsifier is concrete: if future measurements push the baryon ratio outside (6.0, 6.2) × 10⁻¹⁰ at more than 3σ, the squared form fails. Until then, the two-sided correction is the framework's best algebraic candidate, one that improves on the first-order form without yet explaining why the square is exactly right.

/-- The two-sided corrected prediction is strictly smaller than the
    one-sided corrected prediction (`BaryonHigherOrder.eta_B_corrected`). -/
theorem two_sided_corrected_lt_one_sided :
    eta_B_corrected_two_sided < BaryonHigherOrder.eta_B_corrected := by
  unfold eta_B_corrected_two_sided BaryonHigherOrder.eta_B_corrected
  have h := two_sided_stronger_than_one_sided
  have hpos := eta_B_phi_scale_pos
  have := mul_lt_mul_of_pos_right h hpos
  linarith [this]

/-- The corrected RS prediction lies in the band (6.0, 6.2) × 10⁻¹⁰,
    which contains the Planck 2018 central value 6.10 × 10⁻¹⁰. -/
theorem eta_B_corrected_in_observed_band :
    eta_B_corrected_two_sided > 6.0e-10 ∧ eta_B_corrected_two_sided < 6.2e-10 :=
  ⟨eta_B_corrected_lower, eta_B_corrected_upper⟩

/-- The prefactor is order-one: 0 < c_RS < 1. -/
theorem c_RS_in_unit_interval : 0 < c_RS ∧ c_RS < 1 :=
  ⟨c_RS_pos, c_RS_lt_one⟩

What this page does not claim

The physical origin of the square is not derived; it is a hypothesis. The theorem does not include the dynamical factors J_CP, Γ_sph/H, or 1/g★. The agreement with the Planck value is not a proof that the framework's cosmology is correct.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/EtaBPrefactorDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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